A coefficient problem with typically real extremal function

Abstract:

Let $\Sigma _{0}$ denote the class of univalent functions in $|z| > 1$, with expansion $f(z) = z + \sum \nolimits _{n = 1}^\infty {{b_n}{z^{ - n}}}$. We show that if the omitted set of an $f(z) \in {\sum _0}$ is on the trajectory arcs of the quadratic differential $- w(w - \lambda )d{w^2}$ with $\lambda \geqq 4(\sqrt 2 - 1)$, then $f(z)$ has real coefficients. From this we can derive the coefficient estimate of ${\max _{{\Sigma _{0}}}} \mathcal {R}e( - {b_3} - \frac {1}{2}b_1^2 + \lambda {b_2})$.